What to teach in a lecture about diagonalization of matrices in 50 minutes?

I have to give a lecture of 50 minutes with the theme "diagonalization of matrices" (basic undergrad level) for a public contest. I would like suggestions about the contents of this class. I've been thinking to give the pages 181-187 from Hoffman and Kunze's linear algebra book (maybe I will not have time to do all these pages).

What do you think is the best approach? I don't know if I start this class with eigenvalues as Hoffman and Kunze do and goes through very basic theorems about eigenvalues and diagonalizable matrices or begin with the definition of diagonalizable matrices and prove deeper theorems.

• I think it depends on the audience. Are they already familiar with eigenvalues, eigenvectors, algebraic and geometric multiplicities, etc? – Dave Nov 24 '17 at 3:18
• @Dave I'm gonna give this lecture to professors for a public contest. The audience is as like I'm gonna teach basic undergraduate students. I may choose in my lecture if I'm gonna teach students who know eigenvectors, algebraic and geometric multiplicities or not. I don't know which one is more appropriate in this case. – user42912 Nov 24 '17 at 3:25
• I think that given the time constraints, it would be easier if you didn't have to rigorously go over all of these things (eigenvalues, etc.). Maybe you could start by briefly going over them (more as a refresher), but I think the lecture should primarily focus on diagonalization itself. Giving the characterisation of diagonalization (i.e. you can diagonalize a matrix if and only if the algebraic multiplicities equal the corresponding geometric multiplicities in each eigenvalue) is probably the most important part (I would think). Then definitely some examples illustrating this would be good. – Dave Nov 24 '17 at 3:30
• @Dave Thank you for your comments. Do you mean I should focus on the tools to prove the theorem: "A matrix is diagonalizable if and only if the algebraic multiplicities equal the corresponding geometric multiplicities in each eigenvalue."? – user42912 Nov 24 '17 at 3:35
• Yes, I would think that this is one of the most interesting thing about diagonalization to an undergrad class, as this tells us precisely when we can diagonalize a matrix. So I would focus a lot of the time on whatever is needed to prove this theorem, and then state and prove this theorem. Like I also said, I think examples are key to understanding mathematical concepts. So some examples illustrating how to diagonalize a matrix would be good (probably using this theorem; i.e. when can we diagonalize a matrix, and when can't we). Also, maybe explain why diagonal bases are nice. – Dave Nov 24 '17 at 3:44